Optimal. Leaf size=86 \[ -\frac{\left (2-3 x^2\right ) \left (x^4+5\right )^{3/2}}{4 x^4}-\frac{3 \left (15-2 x^2\right ) \sqrt{x^4+5}}{4 x^2}+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right ) \]
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Rubi [A] time = 0.0773522, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1252, 813, 844, 215, 266, 63, 207} \[ -\frac{\left (2-3 x^2\right ) \left (x^4+5\right )^{3/2}}{4 x^4}-\frac{3 \left (15-2 x^2\right ) \sqrt{x^4+5}}{4 x^2}+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Rule 1252
Rule 813
Rule 844
Rule 215
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \left (5+x^4\right )^{3/2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \left (5+x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{(-60-8 x) \sqrt{5+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 \left (15-2 x^2\right ) \sqrt{5+x^4}}{4 x^2}-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{80+120 x}{x \sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{3 \left (15-2 x^2\right ) \sqrt{5+x^4}}{4 x^2}-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac{15}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{5+x^2}} \, dx,x,x^2\right )+\frac{45}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{3 \left (15-2 x^2\right ) \sqrt{5+x^4}}{4 x^2}-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{15}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{5+x}} \, dx,x,x^4\right )\\ &=-\frac{3 \left (15-2 x^2\right ) \sqrt{5+x^4}}{4 x^2}-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{15}{2} \operatorname{Subst}\left (\int \frac{1}{-5+x^2} \, dx,x,\sqrt{5+x^4}\right )\\ &=-\frac{3 \left (15-2 x^2\right ) \sqrt{5+x^4}}{4 x^2}-\frac{\left (2-3 x^2\right ) \left (5+x^4\right )^{3/2}}{4 x^4}+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{5+x^4}}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [C] time = 0.0317762, size = 60, normalized size = 0.7 \[ \frac{1}{125} \left (x^4+5\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{x^4}{5}+1\right )-\frac{15 \sqrt{5} \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};-\frac{x^4}{5}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 73, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5}}+{\frac{45}{4}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) }-{\frac{15}{2\,{x}^{2}}\sqrt{{x}^{4}+5}}+\sqrt{{x}^{4}+5}-{\frac{5}{2\,{x}^{4}}\sqrt{{x}^{4}+5}}-{\frac{3\,\sqrt{5}}{2}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42842, size = 166, normalized size = 1.93 \begin{align*} \frac{3}{4} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \sqrt{x^{4} + 5} - \frac{15 \, \sqrt{x^{4} + 5}}{2 \, x^{2}} + \frac{15 \, \sqrt{x^{4} + 5}}{4 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} - \frac{5 \, \sqrt{x^{4} + 5}}{2 \, x^{4}} + \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55938, size = 204, normalized size = 2.37 \begin{align*} \frac{6 \, \sqrt{5} x^{4} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{x^{2}}\right ) - 45 \, x^{4} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) - 30 \, x^{4} +{\left (3 \, x^{6} + 4 \, x^{4} - 30 \, x^{2} - 10\right )} \sqrt{x^{4} + 5}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.5165, size = 133, normalized size = 1.55 \begin{align*} \frac{3 x^{6}}{4 \sqrt{x^{4} + 5}} - \frac{15 x^{2}}{4 \sqrt{x^{4} + 5}} + \sqrt{x^{4} + 5} + \frac{\sqrt{5} \log{\left (x^{4} \right )}}{2} - \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )} - \frac{\sqrt{5} \operatorname{asinh}{\left (\frac{\sqrt{5}}{x^{2}} \right )}}{2} + \frac{45 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} - \frac{5 \sqrt{1 + \frac{5}{x^{4}}}}{2 x^{2}} - \frac{75}{2 x^{2} \sqrt{x^{4} + 5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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